Q. 17 - Calculus

Question

For each pair of functions $u\left(x\right)$ and $v\left(x\right)$ in Exercises 16-18, fill in the blanks to complete each of the following:

(a) $\frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)$=_______

(b) $\int \_\_\_\_dx=u\left(x\right)v\left(x\right)+C$

(c) $\int udv=$______

$u\left(x\right)=\ln x$, $v\left(x\right)=x$

Step-by-Step Solution

Answer

The blank is

Part (a)$1+\ln x$

Part (b)$1+\ln x$

Part (c)$x\ln x-x+C$

Part (a). Step 1. Given information

The given functions are $u\left(x\right)=\ln x$, $v\left(x\right)=x$.

Part (a). Step 2. Evaluate d d x u x v x .

Differentiate the product of the given functions with respect to x by using integration by parts.

$\begin{array}{rcl}\frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)& =& \frac{d}{dx}\left(x\ln x\right)\\ & =& x\frac{d}{dx}\ln x+\ln x\frac{d}{dx}x\\ & =& x\frac{1}{x}+\ln x\left(1\right)\\ & =& 1+\ln x\end{array}$

Part (b). Step 1. Integration

Integrate the obtained expression for  $\frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)$ with respect to x.

$\begin{array}{rcl}\int \frac{d}{dx}\left(u\left(x\right)v\left(x\right)\right)dx& =& \int \left(1+\ln x\right)dx\\ u\left(x\right)v\left(x\right)+C& =& \int \left(1+\ln x\right)dx\\ & & \end{array}$

Part (c). Step 1. Differentiation

Differentiate $v\left(x\right)=x$ with respect to x to obtain the value of dv.

$\begin{array}{rcl}\frac{d}{dx}v\left(x\right)& =& \frac{d}{dx}x\\ \frac{dv}{dx}& =& 1\\ dv& =& dx\end{array}$

Part (c). Step 2. Integration

Substitute the given and obtained values to evaluate $\int udv$.

$\begin{array}{rcl}\int udv& =& \int \ln xdx\\ & =& \int \ln x·1dx\\ & =& \ln x·x-\int \frac{1}{x}xdx\\ & =& x\ln x-x+C\end{array}$